Why does a group of warmer atoms take up more volume than a group of cooler atoms?

Warmer atoms expand. There is also more movement. This is also explained when putting it into terms with solid vs. gas. In a solid, the atoms vibrate, but they can't move very far because there is not a lot of space between them. In a gas, on the other hand, the atoms are spread apart, and they move very fast. Putting it into context with water, the steam (gas) version is hot, and the ice cube (solid) version is cold. Which one moves more freely? The gas.

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